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I would say that potential energy is a property of a system. If you have a system of two opposite charges separated by a distance r, then there is potential energy associated with the system (because work had to be done against the attractive electrostatic force to separate them by that distance).

This potential energy is a single number. However, if the magnitude of one of the charges changes or the distance between them changes, so too does the potential energy of the overall system. (The number changes, because the system being described has changed.)

Electric potential gives us a way of characterizing the effect associated with ONE single charge without making direct reference to a specific second one. If you look at the definition, you will see that the potential due to a charge is defined as the potential energy that the system*would* have if another positive *unit* test charge *were* located a distance r away from the charge in question. Therefore, electric potential is a function of position r, in space as opposed to being a single number. In fact, potential r, is a scalar field (a quantity that takes on a scalar value at every point in space). The electric potential of a single charge can be thought of as a quantity that characterizes the influence of this charge on its surroundings, but without having to specify what might be there in those surroundings.

This explanation makes it clear why electric potential has units of "potential energy PER unit of charge." It is the potential energy that the system would have if a unit test charge (1 C) were placed at point r, and therefore is neatly independent of the specific amount of charge present at point r.

This potential energy is a single number. However, if the magnitude of one of the charges changes or the distance between them changes, so too does the potential energy of the overall system. (The number changes, because the system being described has changed.)

Electric potential gives us a way of characterizing the effect associated with ONE single charge without making direct reference to a specific second one. If you look at the definition, you will see that the potential due to a charge is defined as the potential energy that the system

This explanation makes it clear why electric potential has units of "potential energy PER unit of charge." It is the potential energy that the system would have if a unit test charge (1 C) were placed at point r, and therefore is neatly independent of the specific amount of charge present at point r.

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The Earth approximates an infinitely large flat plate if the heights involved are relatively small where gravity is constant for these relatively small heights. In this case, gravitational potential is simply the height above (or below) some reference height, for example, define height = 0 at sea level.

Given that gravitiational acceleration is = 9.8 meters / second^2 at all relatively small heights, and and 1 Newton = 1 kilogram meter / second^2, then at height = 30 meters, gravitational potential is 30 meters x 9.8 Newtons / kilogram of mass = 294 joules / kilogram.

A 1 micro gram mass and a 1 kilo gram mass at height 30 meters have the same gravitational potential, but not the same gravitational potential energy (GPE). In this case the 1 microgram mass would have a GPE of 294 x 10^{-9} joules, while the 1 kilogram mass would have a GPE of 294 joules.

I did some of the math in this thread about voltage (electrical potential).

https://www.physicsforums.com/showthread.php?p=2249109

Given that gravitiational acceleration is = 9.8 meters / second^2 at all relatively small heights, and and 1 Newton = 1 kilogram meter / second^2, then at height = 30 meters, gravitational potential is 30 meters x 9.8 Newtons / kilogram of mass = 294 joules / kilogram.

A 1 micro gram mass and a 1 kilo gram mass at height 30 meters have the same gravitational potential, but not the same gravitational potential energy (GPE). In this case the 1 microgram mass would have a GPE of 294 x 10

I did some of the math in this thread about voltage (electrical potential).

https://www.physicsforums.com/showthread.php?p=2249109

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Imagine an empty space with a positively charged ball. This ball creates an electric field all throughout space. The value of that field at any point is sometimes called the potential or the voltage. The potential at any point in space depends on the position and charge of all the objects in your system.

Pick two points around the ball. Take a second charged ball and put it at point A, initially at rest. It will start to move. When the second ball reaches point B, it has picked up some amount of kinetic energy. Where did that energy come from? The answer is while it was at rest at point A, the ball actually had potential energy.

How much potential energy does the second ball have? The answer depends on three values:

* The potential at point A

* The potential at point B

* The charge on the second ball

Most importantly, if we double the charge on the second ball, we double the energy. It's a linear relationship. We can just calculate the energy for one coulomb. That value is now the "exchange rate" between energy and charge. If you have a 5V battery and you want to know how much energy you get when you move some electrons from one terminal to the other, you can figure out that amount by counting the total charge on the electrons, then multiplying it by the exchange rate, 5 joules per coulomb.

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Hi guitarman!

From the PF Library on potential energy …

Electric potential (or electric potential difference or voltage) is confusingly so named, since it is potential energy

Similarly, gravitational potential is potential energy

Unfortunately, the same letter [itex]\bold{F}[/itex] is often used for both "field" and "force", which obscures the fact that (for an "inline" field) the force vector equals the field vector times a charge (for example, electric charge or mass).

Generally, the potential of a vector field [itex]\bold{F}_{field}[/itex] is a scalar function [itex]U[/itex] such that [itex]\bold{F}_{field}\ = \ -\bold{\nabla}U[/itex]. So the potential energy is potential times charge:

[tex]-\int\bold{F}_{force}\cdot\bold{ds}\ = -\ q\int\bold{F}_{field}\cdot\bold{ds}\ = \ q\int (\bold{\nabla}U)\cdot\bold{ds}\ =\ q\int\frac{\partial U}{\partial s}ds\ =\ qU[/itex]

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